Quantum Dimension and Quantum Projective Spaces

TL;DR

This paper reinterprets spectral triples for quantum projective spaces as modular spectral triples, aligning their spectral dimension with classical dimensions and linking to quantum group theory's quantum dimension concept.

Contribution

It introduces a modular perspective to existing spectral triples, reconciling spectral dimension with classical space dimensions and connecting to quantum group theory.

Findings

Spectral dimension matches classical projective space dimension.

Reinterpretation as modular spectral triples.

Connection established with quantum group quantum dimension.

Abstract

We show that the family of spectral triples for quantum projective spaces introduced by D'Andrea and Dabrowski, which have spectral dimension equal to zero, can be reconsidered as modular spectral triples by taking into account the action of the element $K_{2\rho}$ or its inverse. The spectral dimension computed in this sense coincides with the dimension of the classical projective spaces. The connection with the well known notion of quantum dimension of quantum group theory is pointed out.